Face Numbers of Centrally Symmetric Polytopes Produced from Split Graphs

Abstract

We analyze a remarkable class of centrally symmetric polytopes, the Hansen polytopes of split graphs. We confirm Kalai's 3 d -conjecture for such polytopes (they all have at least 3 d nonempty faces) and show that the Hanner polytopes among them (which have exactly 3 d nonempty faces) correspond to threshold graphs. Our study produces a new family of Hansen polytopes that have only 3 d + 16 nonempty faces.

Reflexive polytopes arising from perfect graphs

“…It would, of course, be of interest to find a complete characterization for Ω(Q G 1 , Q G 2 ) to be reflexive. For a finite simple graph G on [d], Ω(Q G , Q G ) is called the Hansen polytope H(G) of G. This polytope possesses nice properties (e.g., centrally symmetric and 2-level) and is studied in [6,22]. Especially, in [6], it is shown that if G is perfect, then H(G) is reflexive.…”

“…For a finite simple graph G on [d], Ω(Q G , Q G ) is called the Hansen polytope H(G) of G. This polytope possesses nice properties (e.g., centrally symmetric and 2-level) and is studied in [6,22]. Especially, in [6], it is shown that if G is perfect, then H(G) is reflexive. Theorem 1.1 (b) says that G is perfect if and only if the Hansen polytope H(G) possesses the integer decomposition property.…”

“…This polytope possesses nice properties (e.g., centrally symmetric and 2-level) and is studied in [6,22]. Especially, in [6], it is shown that if G is perfect, then H(G) is reflexive.…”

$f$-vectors of $3$-polytopes symmetric under rotations and rotary reflections

“…The set of f -vectors of four dimensional polytopes has been studied in [BR73], [Bar74], [AS85], [Bay87], [EKZ03], [Zie02] and [BZ18]. Some insights about f -vectors of centrally symmetric polytopes are given in [BL82], [AN06], [Sta87] and [FHSZ13]. It is still an open question, even in three dimensions, what the f -vectors of symmetric polytopes are.…”

Many 2-Level Polytopes from Matroids

“…It is easy to see that the 2-levelness is preserved for some polytopal constructions: pyramid, prism, and Cartesian product. Moreover some subfamilies of 2-level polytopes are known: two of them are explored in [FHSZ13], the so-called Hansen polytopes [Han77] and Hanner polytopes [Han56], while a third one arises from stable sets of perfect graphs as explained in [GLS93,Ch. 9].…”